Fig 1.
Schematic representation of network architecture, neuronal integration and spike, synaptic conductance traces.
(A) The local recurrent neuronal network consists of excitatory (Exc) and inhibitory (Inh) spiking neurons with synaptic connections (blue, excitatory; red, inhibitory) and inputs from other neural circuits or external stimuli. (B) The voltage trace of one IF neuron with refractory period and leaky current. (C) The unitary conductance response to a pre-synaptic spike is described by a bi-exponential function with latency τl, rise time τr and decay time τd.
Fig 2.
Multi-scale dynamics of E-I balanced network with various synchrony degree.
Left panel: asynchronous state (τd_e = 6 ms, τd_i = 6 ms); Middle panel: moderately synchronized state (τd_e = 4 ms, τd_i = 10 ms); Right panel: highly synchronized state (τd_e = 2 ms, τd_i = 14 ms). (A, C, E) Time series of membrane potential, input conductances, and input currents of a randomly selected neuron. (B, D, F) Network activity. Top, raster plot of a subset 500 neurons (Exc 400 (blue), Inh 100 (red)); bottom, the average excitatory and inhibitory population activity in 1-ms bins; inset, autocorrelation (AC) of the excitatory population activity. Middle and right panels show that the population rhythm is mainly determined by inhibitory decay time τd_i, and the delayed negative feedback from inhibitory population suppresses the firing of the excitatory population, leaving a window for integration, whose size controls the burst of individual activities (C, E).
Fig 3.
Co-existence of multi-scale cortical activities at moderately synchronized states.
(A) Average pairwise 1-ms synchrony between excitatory neurons (E—E Synchrony); (B) Average CV (standard deviation/mean) of the inter-spike intervals (ISIs) over the excitatory population; (C) Power spectra of population activity for 3 different parameter sets indicated in (F); (D) Peak frequency; (E) Peak power. (F) Avalanche size distributions for 3 different parameter sets. (G) Distance of avalanche size distribution from the best-fitted power-law distribution; (H) ISI CV (red), distance from power-law (black) and peak power (blue) vs. E—E Synchrony, showing the co-existence of irregular firing, synchronized oscillations and neuronal avalanches at moderately synchronized states. (A, B, D, E, G) in the parameter space (τd_e, τd_i) (unit: ms).
Fig 4.
Effect of relative resting energy r on optimized energy efficiency ηopt(ρ).
(A, B) The optimized energy efficiency ηopt(ρ) vs. activity level ρ for various values of relative resting energy r in both binary (A) and analog scenarios (B). Larger r shifts the value of ρm for maximal ηopt(ρ) monotonically from ρm → 0 at r = 0 to ρm = 0.5 in binary scenario (open circles in (A)) or ρm = 1 in analog scenario (solid points in (B)) at r → ∞. (C) The monotonic dependence of ρm as well as its corresponding firing rate v (v = ρ/Δτ, Δτ = 20 ms) on r in both binary (black dashed line) and analog scenarios (black solid line). To achieve the maximal energy efficiency ηopt(ρ), the neuronal firing rate is constrained in the range of 1 ∼ 8 Hz for binary patterns (red dashed line) or 1 ∼ 10 Hz for analog patterns (red solid line) with r in the empirical range 0.005 ∼ 0.1, respectively.
Fig 5.
Definition of spatiotemporal spike patterns.
(A) Examples of cross-correlogram between neuron pairs for various parameter sets (τd_e, τd_i) show that spike coincidence happens within 20-ms windows; the average firing rate of one neuron is plotted relative to the time at which the other neuron spikes, averaged over 2000 pairs of randomly selected excitatory neurons. Black, blue, red points are the respective subcritical, critical supercritical cases as exampled in Fig 2. Three more cases around the critical region are shown as green points. (B) Schematics of mapping spiking patterns of 10 randomly selected neurons into binary strings; black, patterns without any spike; blue, binary patterns with spikes.
Fig 6.
Cost-efficient information capacity in the critical region.
(A) Average excitatory firing rate vE; (B) Energy efficiency ηsim in analog scenario at r = 0; (C, D) Energy efficiency ηsim at various r (colors) and average excitatory firing rate vE (black) vs. E—E Synchrony in both binary (C) and analog (D) scenarios. Cost-efficiency is achieved robustly in the critical region across the empirical range of r. n = 40 for all patterns. (A, B) in the parameter space (τd_e, τd_i) (unit: ms).
Fig 7.
Probability of empty patterns.
Dependence of the probability p0 of empty patterns on the number of spiking neurons mn for and the subcritical state in our simulations at various sample size n. Dashed line represents the ideal case with all neurons firing randomly. Parameter set (τd_e, τd_i) is indicated by the triangle in Fig 6A and 6B.
Fig 8.
Trade-off in energy efficiency reduction.
(A) Energy efficiency reduction by burst and synchronization in binary scenario; (B) Energy efficiency reduction is minimal in the critical region in both binary and analog scenarios; r = 0 for (A, B). (C) Comparison of simulated energy efficiency ηsim with the upper bound ηopt at various states for various r. The optimum is represented by solid lines and the simulated by symbols. (D) ηopt − ηsim vs. r in both binary (open circles) and analog (solid points) scenarios. Energy efficiency reduction keeps minimal in the critical region at various r in both scenarios. (C, D) Parameters (τd_e, τd_i) are indicated in Fig 6A and 6B with corresponding symbols; n = 40 for all patterns.
Fig 9.
Spiking neuron number distribution.
Probability distributions of the activated neuron number for the selected states, indicated in Fig 6A and 6B with the corresponding symbols. n = 40 for all patterns. The distribution in the critical region is close to the experimental data [48] (red).